Download Lesson Options and Reflections

Yale Education Studies: Lesson Plan Section 4.6
Teacher: William McKinney
Lesson Topic: Isosceles, Equilateral, and Right s
Date: 12/1/08
Length of class: 40 minutes
Course: Honors Geometry
Planning
I.
Assumptions about the lesson, content, or students:
1. Andrea taught the last two lessons and I’m not positive of how far she got or
where she left off. I’ll assume she finished going over 2-column proofs and that
the students have had practice proving various things. I will also assume that the
students have gone through the book to create a list of the most important
theorems, postulates, properties, and definitions, particularly vertical angles, the
reflexive property, the transitive property, and CPCTC (definition of congruent
triangles).
2. Students may have slight trouble with the first proof if Andrea didn’t get to any
part of Sect 4.6. If she didn’t, then I may have to give a hint (I’ll ask a student if
they started sect 4.6 when they start walking into class). The hint will simply be
written on the left side of the board as if it were being put on the board as new
notes. I will leave the theorem up till we begin our new notes. This is where we
will start our new notes.
II.
Learning Objectives:
1. By the end of the lesson, students should be able to complete their homework
by determining angle and side measurements given the triangles are isosceles
or equilateral.
2. By the end of the lesson, students should be able to determine whether there is
sufficient information to glean whether two right triangles are congruent using
the HL Congruence Thm.
III.
Specific Design:
When the students walk in, a do now will be written on a transparency and projected
on the board. Students were to have learned to write a 2-column proof right before
break, so this will be used as a refresher.
IV.
Link to Standards:
NCTM Content
□ Number & Operations
 Algebra
 Geometry
 Measurement
□ Data Analysis & Probability
Specific Strand:
V.
Assignment due:
N/A
Specific Lesson Development
NCTM Process
□ Problem Solving
 Reasoning & Proof
□ Communication
 Connections
□ Representation
Connecticut & Other
 Algebraic Reasoning
□ Numeric & Proportional Reas.
 Geometry & Measurement
□ Working With Data
□ _________________
1.
Opening Activity or Guided Inquiry
Time Allotted
The first group to prove either of the following will receive 5 bonus points on the
next pop quiz. Any group that can prove both correctly in 10 minutes will receive an
automatic 100% on the next pop quiz.
A
Given BD  EC , AC  AD
Prove AB  AE








BD  EC
AC  AD
1  2
 ADB  ACE
 AB  AE
1
B
E
D
C
Given
Given
Base Angles Theorem
SAS Postulate
CPCTC
X
 Given
 XY || WZ , XZ || WY
Prove X  W



2
Corresponding
Parts of
Corresponding
Triangles are
Congruent
Y

Z



XY || WZ
YZW  XYZ
XZ || WY
 XZY  ZYW
ZY  ZY
 XYZ  WZY
X  W
W
Given
Alternate Interior Angles Theorem
Given
Alternate Interior Angles Theorem
Reflexive Property
ASA Postulate
CPCTC

Students will have the first 10 minutes of class to complete the proofs before we go
over them. If any of the groups were able to complete the proofs, then I will have
them come to the board to solve. There are a couple ways to prove these, so
students may have solutions different to my own.
2.
Grounding the class
After the Do Now, I will ask the students how their Thanksgivings were and if
anyone went anywhere interesting. I will then ask the students to please take out
their notebooks and label their notes for today Sect 4.6-Isosceles, Equilateral, and
Right Triangles. Today’s date is December 1 (I will write today’s date in Chinese on
the board). I will also have extra table of contents for any students that may need
one.
3.
Lesson Outline
9:05 AM
So today we’re going to look at isosceles and right triangles just a little more in
depth, but like always, we’re just going to build on stuff you may have already been
taught and formalize it into mathematical language.
Our first theorem is the Base Angles Theorem, which states
Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are
congruent.
i.e. If AB  AC , then B  C.
Now that we know how to create proofs, why don’t we practice by proving the Base
Angles Thm? So what do we know? i.e. what are we given? Well, we know we’re
 
looking at a triangle and we know that two of the sides are congruent. We’ll go
through the proof together. This will be good practice for them. If they need help, I
will alternate between giving them the statement and giving them the reason.
Ideally, they’ll be able to do the proof without me helping at all, but I’ll assume for
now that they’ll need some help, especially coming back from Thanksgiving.
A
B


Statement
1. ABC
2. AB  AC
3. AD bisects BAC
4. BAD  CAD
5. AD  AD
6. BAD  CAD
7. B  C
D
C
Reason
1. Given
2. Definition of isosceles 
3. By construction
4. Definition of bisector
5. Reflexive Property
6. SAS Postulate
7. CPCTC


Our next theorem is the opposite of the base angles theorem and says
Converse of the Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are
congruent.
i.e. If B  C, then AB  AC .
So the first two theorems had to do with isosceles triangles and help us to better
define what they are. Before, we said isosceles triangles were triangles where two
 
sides were congruent. Now we know that the angles opposite the congruent sides
are also congruent.
The next two corollaries are about equilateral triangles. Who can remind me what
an equilateral triangle is? How do we define it right now? Just by the sides, right?
Well, our first corollary is an add on to the Base Angles Thm. If we know that two
sides are congruent, then we know the two opposite angles are congruent. So let’s
look at all the pairings of congruent sides. What ends up being true? It turns out that
the equilateral triangle is just a special case of the isosceles triangle where all the
sides are congruent and all the angles are congruent. So let’s formalize this.
Corollary to the Base s Thm:
If a  is equilateral, then it is equiangular.
The second corollary is the opposite and follows from the Converse to the Thm.
Corollary to the Converse of the Base  Thm:
If a  is equiangular, then it is equilateral.
So let’s look at an example so we can practice what we’ve just learned. Use the
diagram to find the values of x and y.
y
x
So where do we start? Well, what do we know about equilateral triangles? All the
angles are congruent. And how many degrees does a triangle have? 180 o. So how
much is each angle? 180/3 = 60o. So let’s label our diagram (remember, labeling
the diagram often makes solving the problem and making connections easier. So
don’t skip this step). Does anyone have an idea of how we can find the vertex angle
of the second triangle? What’s its relationship to the bottom angle on the right?
They form a linear pair! So that means the pair must what? Add up to 180o. So the
vertex angle is 180 – 60 = 120o. Now the second angle is an isosceles angle, so we
know the two base angles are equal and must add to 180 – 120 = 60o. And 60/2 =
30o.
So x = 60o and y = 30o.
Now I want you to try one on your own. Everyone take two minutes to solve for x
using the following information/diagram. If you have your books, it’s problem 19 on
page 240. When you finish, check your answer in the back of the book. We’ll go
over it together after everyone’s had a chance to try and solve it.
56 ft
8x ft
So all the sides of the equiangular triangle are 56 ft by the Corollary to the Converse
of the Base Angle Thm. And by the definition of isosceles triangle, 56 ft = 8x. So x =
7.
Finally, we have one last theorem, and it has to do with right triangles. It follows
Hypotenuse-Leg (HL) Congruence Theorem
If the hypotenuse and a leg of a right  are congruent to the hypotenuse and
a leg of a second right , then the two triangles are congruent.
So let’s look at the picture. Guess what, you’re finally getting your SSA Thm to show
two triangles are congruent. But REMEMBER, it only works on right triangles! This
is why we will call it the HL Congruence Thm, so we don’t accidentally forget and
apply it to all triangles.
4.
Closure
I will now assign the homework.
5.
Assignment for next class
p239 #8-22 (evens) (8 problems)
Lesson Options and Reflections
6.
Exit Ramps
If there is time, I will also have the students look at p239 #5.
If students have already completed notes on Sect 4.6, then we will move to Sect
4.7. We will still start with the same Do Now. The only difference will be the new
notes.
7.
Plan B idea(s)
This is a potentially a Plan B if the students have already gone over Sect 4.6 with
Andrea. If they are still having trouble, then we may go over the lesson again since
we have time and it will solidify the concepts before we move on to the final section.
8.
Self-Evaluation